Dual Regularization and Outer Approximation of Optimal Control Problems in BV

TL;DR

This paper develops a regularization and outer approximation approach for elliptic optimal control problems with total variation constraints, proving convergence and deriving optimality conditions.

Contribution

It introduces a quadratic regularization of the dual TV representation and an outer approximation algorithm, with proven convergence and optimality conditions for the original problem.

Findings

Convergence of the regularized problem as the regularization parameter vanishes.

Convergence of the outer approximation algorithm.

Numerical experiments confirming theoretical results.

Abstract

This paper is concerned with an elliptic optimal control problem with total variation (TV) restriction on the control in the constraints. We introduce a regularized optimal control problem by applying a quadratic regularization of the dual representation of the TV-seminorm. The regularized optimal control problem can be solved by means of an outer approximation algorithm. Convergence of the regularization for vanishing regularization parameter as well as convergence of the outer approximation algorithm is proven. Moreover, we derive necessary and sufficient optimality conditions for the original unregularized optimal control problem and use these to construct an exact solution that we use in our numerical experiments to confirm our theoretical results.